756 Mr. H. D. Arnold on Stokes's Law/or 



object of this paper to discuss the departures from Stokes's 

 law due to the velocity of the spheres and to slip at the 

 interface of sphere and liquid. 



The force which opposes the steady motion of a sphere 

 through a viscous liquid as deduced by Stokes* on purely 

 theoretical grounds is given by 



F = 67T//,?'V, 



where fi is the coefficient of viscosity of the fluid, r the 

 radius of the sphere, and v its velocity. From this it follows 

 that if a sphere of density a is moving in a liquid of density 

 p and is acted on by no external force except gravity, since 

 its motion will become steady when the impressed force is 

 balanced by the viscous resistance, that is when 



4 



o7rr 3 (cr — p) g=67T purv, 



o 



its terminal velocity will be given by 



2 2 o— p 



y pu 



In the mathematical derivation of the original formula 

 certain assumptions are made, notably 



1. That the discontinuities of the fluid are small compared 

 with the size ofthe sphere. 



2. That the fluid is infinite in extent. 



3. That the sphere is smooth and rigid. 



4. That there is no slip at the surface between sphere and 

 fluid. 



5. That the velocity of the sphere is small. 

 Cunningham f has investigated theoretically, and Millikan J 



experimentally, the effect of a violation ofthe first assumption, 

 and they have shown that in the case of a very small sphere 

 falling in a gas, there is a considerable departure from the 

 value given by the original formula. In the present investi- 

 gation, however, we are dealing only with liquids, and with 

 spheres of comparatively large size, so that we may consider 

 this assumption as sufficiently justified. 



The second assumption has been investigated both theo- 

 retically and experimentally by Ladenberg §, who found that 



* G. G. Stokes, Mathmatical and Physical Papers, iii. p. 59. 



+ Cunningham, Proc. Hoy. Soc. ser. A, lxxxiii. pp. 3-57-3<J5 (March 

 1910). 



X Phys. Rev. April 1911. 



§ R. Ladenberg-, Ann. der Phys. xxii. p. 287 (1907), and xxiii. p. 447 

 (1907). 



